Physics mechanisms underlying the optimization of coherent heat transfer across width-modulated nanowaveguides with calculations and machine learning

Optimization of heat transfer at the nanoscale is necessary for efficient modern technology applications in nanoelectronics, energy conversion, and quantum technologies. In such applications, phonons dominate thermal transport and optimal performance requires minimum phonon conduction. Coherent phonon conduction is minimized by maximum disorder in the aperiodic modulation profile of width-modulated nanowaveguides, according to a physics rule. It is minimized for moderate disorder against physics intuition in composite nanostructures. Such counter behaviors call for a better understanding of the optimization of phonon transport in non-uniform nanostructures. We have explored mechanisms underlying the optimization of width-modulated nanowaveguides with calculations and machine learning, and we report on generic behavior. We show that the distribution of the thermal conductance among the aperiodic width-modulation configurations is controlled by the modulation degree irrespective of choices of constituent material, width-modulation-geometry, and composition constraints. The efficiency of Bayesian optimization is evaluated against increasing temperature and sample size. It is found that it decreases with increasing temperature due to thermal broadening of the thermal conductance distribution. It shows weak dependence on temperature in samples with high discreteness in the distribution spectrum. Our work provides new physics insight and indicates research pathways to optimize heat transfer in non-uniform nanostructures.


Introduction
Lattice heat conduction is the major contribution to the parasitic heat flow across nanostructures of semiconductors.It must be controlled in many applications, such as highdensity nanoelectronics circuits, sensing, thermal insulation, and thermoelectric energy conversion.Thermoelectric generators (TEGs) are devices that convert thermal energy into electric power with no moving parts, they are highly reliable and have long lifetimes.Techno-economic analysis showed that TEGs have the full potential to compete with conventional power sources and power the internet of things [1].They can also serve as thermoelectric coolers (TECs) and cope with the problem of overheating of nanoelectronic circuits.Micro-TEGs and micro-TECs ideally serve these purposes because they are compatible and can be integrated on microelectronic boards.Thermoelectric metamaterials such as width-modulated nanowaveguides (nWVGs) would be suitable for commercial thermoelectric applications if optimized for minimum thermal conduction and maximum thermoelectric efficiency [2].Width modulation has been recently optimized for minimum thermal conduction [3].This work aims to shed light on the physics mechanisms underlying the optimization and indicate design pathways to control heat transfer across nWVGs.
Quantum confinement and nanoscale periodicity modify phonon dispersion and affect the material's thermal, optical, electrical, and mechanical properties [4][5][6][7][8][9].These effects drastically limit parasitic coherent heat conduction and remarkably enhance thermoelectric energy conversion efficiency [10][11][12][13] in metamaterials with high-quality interfaces and boundaries and characteristic dimensions shorter than the dominant phonon wavelength [14][15][16].Recent studies on thermal conduction in metamaterials such as superlattices, nanomeshes, holey nanobeams, and width-modulated nWVGs confirm the dominance of coherent phonon transport at low temperatures [17,18] and at elevated temperatures [19][20][21][22][23][24][25][26][27][28].Coherent phonon transport can be geometrically tuned in width-modulated nWVGs by designing aperiodicity in the modulation profile [29][30][31].Optimization of geometrical aperiodicity can be addressed with the aid of machine learning (ML) because of the large number of aperiodic configurations and the high degree of complexity.ML has been rapidly established as the 'fourth paradigm for scientific research' that could complement the first three paradigms, theory, experiment, and simulation [32].It can be ideally combined with physics models and calculations to accelerate the understanding, selection, and design of materials and their structures [33][34][35][36][37][38][39].This is particularly important in dealing with problems with a high degree of complexity, competing mechanisms, and interdependencies.Such problems are very demanding and, often, cannot be addressed at an affordable cost by the other paradigms.Thermal conduction in the presence of geometrical nanoscale non-uniformity is a representative example.In the last years, there has been an explosion of machine-learning-assisted research activity in thermal science.This involves material selection [40][41][42][43] and nanostructure design for optimal thermal properties [44][45][46].Recently, efficient Bayesian optimization of geometrical aperiodicity for minimum coherent thermal conduction was demonstrated in width-modulated nWVGs [3].Calculations and ML showed that aperiodic width modulation is optimized for minimum thermal conductance by maximum disorder in the modulation profile of nWVGs according to a physics rule.On the other hand, optimal aperiodicity was found for moderate disorder in the case of heterostructures against physics intuition [44,[47][48][49][50][51][52].Such counterevidence indicates the non-trivial role of disorder in minimizing lattice heat conduction and the need for further understanding of the physical processes underlying the optimization of non-uniform nanostructures and stimulated the present work.
The main objective of this paper is to provide physics insight into the role of disorder in decreasing thermal conductance and elucidate why thermal conductance is minimized for maximum disorder in width-modulated nWVGs.For this, calculations on full sets of nWVGs are analyzed to interpret the distribution of the thermal conductance among the aperiodic configurations.It is revealed that the thermal conductance values are ordered in zones with an increasing degree of disorder in the modulation profile of the corresponding configurations.Additional objectives are to explore the impact of composition, modulation geometry, and sample size on the thermal conductance distribution and the optimization efficiency.Calculations for nWVGs of different constituent materials and width-modulated geometries of interest [2] show that the dominant mechanism determining the distribution of thermal conductance is the degree of modulation and support the general validity of the physics optimization rule.The present work also explores the effect of thermal broadening on the distribution of thermal conductance, the Bayesian optimization efficiency, and the minimization of thermal conductance in connection with the sample size and the discreteness of the distribution spectrum.The system and the methodology are detailed in section 2. The results are presented and discussed in section 3. Section 4 is devoted to remarks and conclusions.

Methodology
The structures of interest are three-dimensional nWVGs with two-dimensional confinement and one propagation direction.The width of the nWVGs is modulated along the propagation direction (figure 1).
The modulated nWVGs consist of arrays of layers of the same material with different widths and lengths.The arrays can be periodic or aperiodic.Layers are either wide ('openings') or thin ('constrictions').Wide leads are assumed at the two endings of the nWVGS.The optimization objective is to find the optimal array of layers, the one with minimum thermal conductance.This is referred to as the 'optimal modulation profile'.To perform the optimization, we need to fix the material and the number N of layers.Calculations and optimization are discussed for two reference materials, GaAs and Si,  COMBO [57,58] and two width-modulation geometries (figure 1(b)).Thermal conductance was calculated within elastic wave transmission theory [53,54] (table 1) as detailed in [30,31,55,56].The dimensions and material properties are continuous parameters.Calculations are shown for nWVGs with constriction layers width of 10 nm and opening layers width of 100 nm.The selected dimensions are suitable for experimental realization and applications of technological interest.
The ML optimization was performed using the open-source Bayesian optimization library COMBO [57,58] (table 1), which has been tested for optimizations in various energy transport problems [45][46][47].The objective function is the value of the thermal conductance.The ML algorithm is integrated with the property calculator in a closed-loop iterative optimization (figure 2).The process we followed is illustrated in figure 2.
Optimization requires digital representation of the nWVGs.For this, binary flag numbers are chosen as suitable descriptors to represent the arrays of layers.In the binary representation, the binary flag '0' denotes constriction layers and the binary flag '1' denotes opening layers (figure 2).A quantum dot (QD) modulation unit is defined by a sequence of opening layers n between two constriction endings.Thus, the digital representation of a QD is a sequence of '1 s' surrounded by '0' layers at opposite endings.In this definition, QDs of different sizes are formed by sequences of layers '1' of different lengths, i.e. different numbers of layers.Subsequent layers '0' of different numbers represent constrictions of different lengths.In this representation, the descriptor of a periodic SL with N = 12 layers is 010101010101 and the descriptor for an aperiodic SL with 12 layers is 011101111100.
As illustrated in the flowchart of figure 2, the thermal conductances of n candidates randomly selected are calculated first.These values of thermal conductance and the corresponding n descriptors are then used to train a Bayesian regression function.Next, the thermal conductance values for each of the remaining candidates are estimated by a Bayesian posterior probability distribution derived with a Gaussian process.The acquisition function uses the expected improvement criterion to indicate the best candidate to proceed with.The process continues with the calculation of the thermal conductance of this candidate and the inclusion of its value in the training set.The loop is repeated until the process converges to the candidate with minimum thermal conductance, referred to as the optimal width-modulated nWVG.The efficiency of the optimization is the higher the smaller the number of training candidates until convergence is achieved.

Decrease of thermal conductance and aperiodicity/disorder
The thermal conductance of width-modulated nWVGs depends on the shape of the modulation profile, and the presence of order or disorder [30,31].It is smaller than that of the corresponding uniform nWVG.Even a single constriction  significantly decreases the thermal conductance of coherent phonons.The addition of more constrictions decreases it further down to the periodic superlattice (p-SL) value when the modulation units are identical and below this limit when the modulation units are non-identical (figure 3).
The maximum decrease occurs for the maximum number of non-identical modulation units in the modulation profile [3,30,31].In the quantum confinement regime, the underlying mechanism for the decrease of thermal conductance is the reduction of the phonon transmission probability due to quantum interference between phonon waves scattered at the width-discontinuities of the nWVG.Quantum interference is sensitive to the geometrical arrangement of discontinuities and thus the values of the thermal conductance of the various configurations of the modulation profile are different.Calculations on full sets of width-mismatch modulation profiles showed that the distribution of the thermal conductance among the different configurations has a peaked structure with a long tail and a well-defined minimum (figure 3).To better understand the origin of this distribution and the role of the aperiodicity, we explored the characteristics of configurations in the different parts of the distribution.

Distribution of thermal conductance and degree of disorder
For a modulation length of N layers, there are 2 N configurations.The number of configurations increases rapidly with increasing N; it is 256, 4096, and 16 384 for N = 8,12, and 14 respectively.We consider the configurations for N = 8, the smallest yet representative statistical set.The corresponding distribution histogram is shown in figure 4. The inset shows the distribution of the full set of 256 configurations including the uniform nWVG for which thermal conductance is maximum.
There is a large gap between this maximum and the thermal conductance of any of the modulated nWVGs.The main panel zooms in on the modulated nWVGs, without the uniform nWVG.Thermal conductance is plotted relative to that of the p-SL.The thermal conductance values of the modulated nWVGs appear to be distributed into three zones: zone Athe highest values zone, zone B-the medium values zone, and zone C-the lowest values zone.We analysed the modulation profile of the configurations in each zone to determine each zone's identity.Representative configurations are shown in figure 5. Zone A includes configurations with a single constriction and zero QDs.The constriction length of candidates in this zone ranges from 1 to 8 layers.Zone A is a low-modulationdegree zone.Zones B and C include modulated nWVGs with more than one constriction and at least one QD.Zone B includes periodic and aperiodic configurations.Among the periodic ones is the p-SL 01010101 with relative thermal conductance equal to 1.This is the periodic configuration with the highest modulation degree of four identical QDs (010).Other periodic configurations have profiles modulated by a smaller Imposing the 50% composition constraint to the N = 8 set reduces the number of configurations from 256 to 70.The effect of the composition constraint in the thermal conductance distribution is illustrated in figure 5 where the distribution histograms of the thermal conductance among the candidates of the full and the reduced sets are plotted together.
It can be noticed that although the histogram of the reduced set is sparser, the candidates are still well distributed in the three zones A, B, and C.These zones are now more discrete and even more distinguishable.The thermal conductance decreases monotonically with increasing modulation degree from zones A to C. The optimal configuration for the reduced set follows the physics rule.The statistical distribution has the same characteristics under the composition constraint.
The distribution histogram of the thermal conductance for a larger set of 3432 configurations for N = 14 and 50% composition, is shown in the inset of figure 3.In the case of this larger statistical sample, the distribution is more continuous and shows a single dominant peak instead of sequences of distinct intermediate peaks as in the case of the smaller sample of figure 5. Zone A has one band of configurations with a single constriction of 7 layers.Zones A and B are grouped by a rather smooth envelope to an extended zone with a peak and a long tail.The p-SL lies inside the tail.Analysis of the distribution of candidates shows that thermal conductance decreases gradually with increasing modulation degree.At first, increasing the modulation degree decreases thermal conductance down to the p-SL limit.Thermal conductance decreases below the p-SL value for increasing aperiodicity.Around the peak are distributed configurations with identical and non-identical QDs in their modulation profile, with a medium degree of modulation.The peak belongs to zone B because most configurations have a medium modulation degree.Thermal conductance decreases rapidly with increasing modulation degree after the peak, in zone C. Most configurations in this zone are arrays with a maximum number of non-identical QDs.The optimal configuration with minimum thermal conductance has been identified at the low edge of this zone.Its descriptor is 00110100101110.It is degenerate with its reverse.Its schematic profile is also shown in figure 3. It is composed of four QDs, three of which are non-identical: 010, 0110, and 01110.We found the same optimal configuration when we performed the optimization for a different geometry with a modulation profile symmetric around the nWVG central axis (figure 3).The same optimal configuration was found for GaAs and Si.The optimal configuration is in all cases the one with maximum modulation degree due to the distribution of the thermal conductance among the aperiodic width-modulation configurations: the decrease of thermal conductance is controlled by the modulation degree irrespective of choices of constituent material, width-modulation-geometry, and composition constraints.

Underlying physics mechanism
The generic behavior of the distribution of the thermal conductance is interpreted by the underlying physics mechanism: destructive phonon wave interference at the width-modulation discontinuities that becomes more significant with increasing modulation degree [30].Analysis of the distribution histograms shows that thermal conductance gradually decreases with an increasing degree of modulation, from the singleconstriction value towards the p-SL value, reaching its minimum at the optimal aperiodic array.The gradual decrease corresponds to the gradual perturbation of the step-like transmission coefficient of the uniform nWVG by more extended destructive interference upon increasing modulation degree as illustrated in figure 6.
The perfect propagation steps of the uniform nWVG are slightly distorted by a spectrum of shallow wave interference patterns in the case of a single constriction modulation.The distortion deepens and gets more severe with increasing modulation degree upon increasing the number of non-identical modulation units.Zooming in on the first propagation channel shows that even long wavelength phonons 'see' modulation and their propagation is considerably affected.In the case of a single-constriction modulation, resolved propagation peaks appear first, followed by a continuum of propagation states with fluctuating transmission probability.The modification is enhanced in the case of the p-SL, where distinct propagation zones are formed by destructive interference due to periodicity.Propagation zones get gradually narrower as modulation deviates more from the periodic one.They shrink by extended destructive interference due to the absence of periodicity in the optimal aperiodic configuration.The transmission coefficient for the optimal configuration is smaller than for the single-constriction modulation as well as for the p-SL modulation.Thermal broadening screens effects of interference.This explains the convergence of the three curves of thermal conductance with increasing temperature (figure 3).The effects of thermal broadening on the thermal conductance distribution and the Bayesian optimization efficiency are further discussed in the next subsection.

Thermal broadening and optimization efficiency
Thermal broadening screens quantum confinement effects in transport when temperature increases.The distribution histogram of the thermal conductance broadens with increasing temperature as illustrated in figure 7 where the distribution histograms for the N = 14 configuration set are plotted for three temperatures 2 K, 5 K, and 10 K.
Increasing thermal energy broadens and lowers the peak of the distribution.Such broadening is typical for transport dominated by quantum effects, wave interference in the present case.The peaked structure of the distribution and the welldefined minimum threshold remain clear with increasing temperature.The existence of a minimum thermal conductance at all temperatures confirms that the problem is suitable for optimization in the whole temperature range considered here.The efficiency of the optimization is shown in figure 8 for the three temperatures.
In all cases, we started the optimization by randomly selecting a group of initial candidates out of the pool of 3432 candidates.We then performed rounds of optimization with different sets of group candidates until all content of our pool was used.We repeated the procedure for several different choices of a random initial set of candidates.We show representative results for two initial random sets and rounds of groups of 20 candidates.At 2 K, the optimal configuration was identified after calculating the thermal conductance of 3%-8% of the candidates (minimum of ∼100 candidates).At 5 K, the optimal configuration was identified after calculating the thermal conductance of 7%-30% of the candidates (minimum of ∼250 candidates).At 10 K, the optimal configuration was identified after calculating the thermal conductance of 15%-20% of the candidates (minimum of ∼500 candidates).The efficiency of the optimization drops with increasing temperature.This can be understood by that thermal broadening reduces the spreading of thermal conductance among candidates and candidates with similar degrees of modulation are less easily distinguishable.Convergence is thereby delayed.Remarkably, at all temperatures optimization follows the physics optimization rule from the very early steps, irrespective of the time it takes it to converge as shown in figure 8 where the optimization evolution is detailed for the three temperatures.At all temperatures, optimization evolves through candidates that are arrays of permutations of the same set of non-identical QDs.The optimal configuration is the optimal permutation of the actual set of nonidentical QDs at a given temperature.Notably, the efficiency of the optimization remains high despite the thermal broadening of the distribution.This is attributed to the existence of a well-defined low threshold of the distribution envelope with a steep ending at all temperatures.
Effects of thermal broadening are less pronounced in smaller statistical samples where the distribution is sparser.To make this evident we show in figure 9, the distribution histograms for the N = 12 set of 924 configurations with 50% composition for comparison with the corresponding data in figures 7 and 9 for N = 14.
In the case of the smaller set, the distribution histogram shows discrete peaks at all temperatures.The range of values of thermal conductance increases with increasing temperature but the heights of the peaks do not show the systematic decrease shown in figure 7.At the low threshold of the distribution, the spectrum is characterized by well-resolved peaks at all temperatures.The efficiency of the Bayesian optimization is comparable at the three temperatures indicating that thermal broadening does not decrease in this case the efficiency of the optimization.This is attributed to the discreteness of the distribution spectrum.It can be concluded that the effect of thermal broadening on the efficiency of Bayesian optimization depends on the discreteness of the distribution spectrum.Thermal broadening of quantum transport effects and discreteness of the thermal conductance statistical distribution act in opposite directions in the efficiency of the optimization.

Concluding remarks
This work addresses open questions on the heat transfer across width-modulated nWVGs that are currently considered promising metamaterials for efficient micro-TEGs and micro-TECs.It explores the physics mechanisms underlying the optimization of the thermal conductance of width-modulated nWVGs with calculations and ML.Analysis of the distribution of thermal conductance reveals distinct zones of values where configurations have different degrees of disorder in their modulation profile.These zones are ordered in an increasing degree of disorder.Calculations on samples of different compositions and geometries indicate that this ordering is generic.It is shown that the distribution of the thermal conductance among the aperiodic width-modulation configurations is controlled by their modulation degree irrespective of choices of constituent material, width-modulation geometry, and composition constraints.This implies that the width-modulation degree is the dominant mechanism in the optimization process of width-modulated nWVGs and is responsible for optimization according to physics.The same behavior is expected in other types of nanostructures due to geometrical modulation.Different behavior should though hold in nanostructures due to compositional modulation as in hetero-SLs where optimization is for moderate disorder against physics intuition.Analysis like the present one could reveal the dominant mechanism in the thermal conductance distribution and the optimization of hetero-SLs.
The present work also addresses the effect of thermal broadening on the thermal conductance distribution and the optimization efficiency.The efficiency of the optimization is evaluated against thermal broadening in connection with the discreteness of the distribution of the thermal conductance among the aperiodic configurations.Optimization of samples of different sizes performed at different temperatures shows that the effect of thermal broadening on the efficiency of the optimization depends on the discreteness of the distribution of the thermal conductance among the aperiodic configurations.It decreases with increasing temperature due to the thermal broadening of the distribution.It remains high with increasing temperature when the distribution is characterized by high discreteness.
The outcomes of this study hold for coherent phonon transport.Deviations are to be expected when the assumption of coherency breaks due to phonon scattering at imperfect boundaries, by impurities and/or phonon-phonon scattering that get increasingly important as temperature increases [14].The range of validity of coherent phonon transport in phononic metamaterials is an open question, a matter of ongoing research and scientific debate [17,20].It is still unclear when experimental evidence should be attributed to coherent and when to incoherent phonon transport [18].This work contributes to resolving this issue pointing out characteristic behavior as a signature of coherent phonon transport in geometrymodulated metamaterials.The outcomes are directly relevant to the scientific research in the field of quantum technologies that are of growing importance for our society.It is broadly known that thermal transport effects have a major impact on deteriorating the efficiency of the operation of quantum devices.These devices operate at low temperatures where coherent phonons dominate thermal transport and our outcomes can be reliably used for designing them, controlling heat transfer, and enhancing their efficiency.A challenging future scope of the research is to extend the formalism to accommodate phonon scattering and study the distribution of the thermal conductance among the aperiodic configurations of width-modulated nWVGs, perform the optimization of width-modulation for minimum thermal conduction, and interpret the optimization.This would clarify whether the identified physics optimization rule is also valid for incoherent phonon transport that dominates in applications at elevated temperatures.
In conclusion, this work provides new physics insight that could help to control heat transmission in nanodevices.It elucidates the role of disorder in decreasing thermal conduction in the quantum confinement regime.It opens design pathways to optimize metamaterials for efficient heat management and thermoelectric energy conversion at the nanoscale.

Figure 2 .
Figure 2. Flowchart of the optimization methodology described in the main text.

Figure 3 .
Figure 3. Thermal conductance relative to the periodic superlattice (SL) versus temperature for single constriction (grey), periodic SLs (green), and optimal aperiodic SLs (orange) for two width-modulation geometries.The inset shows the distribution of thermal conductance among the width-modulation configurations.

Figure 5 .
Figure 5. Thermal conductance distribution histogram for N = 8, for the full set of 256 candidates (hollow columns), and the 70 candidates with 50% composition (orange columns).Representative modulation patterns for zones (A), (B), and (C) are also shown.

Figure 6 .
Figure 6.Energy dependence of the transmission coefficient for single-constriction-(grey), periodic-(green), and optimal aperiodic (orange) modulations.The main figure zooms in the first uniform-nWVG subband.

Figure 8 .
Figure 8.The effect of thermal broadening on the performance of the Bayesian optimization for two paths starting with different sets of random candidates (continuous and dashed lines respectively) and for three temperatures, T = 2 K, 5 K, and 10 K.The sequences of candidates for evolution paths at different temperatures are shown.

Figure 9 .
Figure 9. Performance of the Bayesian optimization for sparse thermal conductance distribution for two paths starting with different sets of random candidates (continuous and dashed lines respectively) and for three temperatures, T = 2 K, 5 K, and 10 K.

Table 1 .
Theoretical and numerical approaches.